Mech 473 Lectures
Professor Rodney Herring
Objectives
At the end of this lecture you should be able to:
• Know the theoretical density of materials and how to calculate
it for single element and compound materials.
• Know the thermal properties of materials.
• Discuss the role of atomic binding energy on the physical
properties of materials
Material Properties for Design of Structures
and Components
Physical Properties
Density
• One of the most important properties of a material is its density
because it determines the weight of the structure or component.
• It depends on the mean atomic mass of its atoms, their atomic or
ionic sizes, and their crystal structures (the way they are packed).
• The density of materials primarily arises from their atomic mass
(1 for H and 207 for lead)
• Metals are dense because they consist of heavy atoms, which tend
to be closely packed.
• Polymers are not dense because they primarily consist of carbon
and hydrogen.
• Ceramics tend to have low density because they primarily consist
of C, O and N and have a low packing fraction.
Theoretical Density
• The theoretical density can be calculated using the properties of
the crystal structure.
• The general formula is:
Density 
number of atoms unit cell atomic mass 
volume of unit cell Avogadro' s Number 
• If a material is ionic or consists of different types of atoms or ions,
this formula has to be modified accordingly. We will see some
examples to clarify.
• Avogadro’s number, NA, is the number of atoms or molecules in
one mole.
– NA = 6.02 x 1023 atoms/mole
Crystal Properties - Density
• The calculation of the density is an immediate extension of the
calculation of the packing fraction of a unit cell.
• The mass per atom(s) is given by the atomic mass, i.e., mass/mole
divided by the number of atoms per mole (Avogradro’s number).
• The volume is given by the dimensions of the unit cell.
• Eg., For Indium, the structure is body centered tetragonal with ao =
0.32517 nm and co = 0.49459 nm. The atomic mass = 114.82 g/mol.
Calculate its density.
g 
 atoms 
2
114.82

mass
cell 
mole 

 calculated 

volume 3.2517 10 8 cm 2 4.9459 10 8 cm 6.02 10 23 atoms/mole




 calculated  7.32 g cm 3
 measured  7.286 g cm 3
Why would there be a difference between
the calculated and measured density?
Crystal Properties - Density
• Calculate the density of Magnesium Oxide, MgO, which has the
“NaCl” structure.
• Eg., The atomic weight of Mg = 24.312 g/mol and O is 16
g/mole. For MgO, the ions touch along the edge of the cube in
the FCC NaCl crystal structure so ao = 2r (Mg) + 2r (O) =
2(0.066) + 2(0.132) nm = 0.396 nm. Calculate its density.
g   O ions 
g 
 Mg ions 
4
24
.
312

4
16


 


mass
unit
cell
mole
unit
cell
mole

 


 calculated 

3
volume
3.96 10 8 cm 6.02 10 23 atoms/mole

 calculated  4.31 g cm 3


Physical Properties
The Melting Point
• The bonding energy of the atoms is directly proportional to the
melting point (Tm) of the material.
• The bond energy is due to the long-range attractive energy of
the atoms between positive and negative charges, which is
Coulombic.
• The melting point can be related to the modulus of elasticity, E, of
a material by,
100 kTm
E

Where k is the Boltzman constant and  is the atomic volume
in the structure of the material.
Binding Energy & Interatomic Spacing
Interatomic spacing
• The equilibrium distance between atoms is caused by a balance
between repulsive and attractive forces. Equilibrium separation
occurs when the total-atomic energy of the pair of atoms is at a
minimum, or when no net force is acting to either attract or repel the
atoms.
• The interatomic spacing is approximately equal to the atomic
diameter or, for ionic materials, the sum of the two different ionic
radii.
A material that has a high
binding energy will have a
high melting temperature
and a high strength.
Binding Energy, Interatomic Spacing &
Linear Thermal Expansion
The coefficient of thermal expansion is influenced by the interatomic
bonding.
Thermal Properties
The Melting Point
The thermal properties of a material can be studied in space
using plasma crystals and the special condition of
microgravity.
What is microgravity?
What is a plasma crystal?
Microgravity is used to study the physical properties
and behaviour of materials because …
Microgravity reduces:
1) hydrostatic pressure
2) sedimentation/buoyancy
3) thermal convection
1g
0g
(microgravity)
In this example, diffusion rather than
convection-driven process dominates
the burning of a candle in microgravity.
A quiescent environment is obtained.
Where can we find the microgravity condition? – Anywhere that has
a “free-fall” condition, i.e., a satellite, space shuttle or space station
orbiting the earth in space, in drop shafts , in drop tubes or at times
when a plane flies in a parabola (parabolic flight).
Plasma crystals, which only exist in microgravity, can
be used to show the atomic melting behaviour of
materials.
Plasma crystals are a collection
-
+
Plasma
I-i
+
+
Ie
-
particle
+
rd u s t -
+
+
D
+
of charged dust particles
surrounded by trapped
electrons, which are generated
by irradiation of the particles by
a Laser, hn
+
h
-
+
-
 ~ 104 electrons on a
particle of 5µm diameter.
Experiments using plasma
crystals are being performed
on the International Space
Station.
Complex Plasmas
Wth
-
-
Wpp
-
-
-
-
-
-
-
Coulomb interaction
 strong coupling, if the
attractive work function
(Wpp) is greater than the
repulsive work function
(Wth).
-
-
-
-
-

Wpp
Wth
1
Complex Plasmas
 ~ 0.2 mm
-
-
-
-
-
-
-
-
-
-
-
-
 strong coupling
 Coulomb crystallization
  1
-
-
Coulomb interaction
Plasma Crystal
forms if an ordered
structure of the ions
occurs.
Crystalline System
Pressure: 370µbar
RF-Voltage (Vss): 32V
Particle Size: 3,4µm
Gas: Argon
Volume: 2.3 x 1.7 x 2.3 mm³
Random orientation showing three dimensional distribution of ions
floating in space.
The Melting of a Plasma Crystal
Plasma Crystals Grown in Space
Give insight into physical phenomena such as thermal properties,
which is otherwise unobservable terrestrially.
• Useful to study the thermally-induced transition between solid –
liquid – gas – plasma states of matter.
Later in this course we will use plasma crystal to study the following:
• The formation of defects
• Phase separation (different crystal structures formed by small and
large atoms)
• Convection of fluids at interfaces
• Flow of heat through a crystal via phonons (lattice vibrations)
Thermal Properties
The Melting Point
How does the melting temperature come into play in material design?
• It is a factor to use a material in a high temperature environment,
such as that used in combustion engines and nuclear reactors.
• A high temperature for one material may be a low temperature with
respect to another due to a difference in melting temperatures.
• To determine the effect of temperature on materials properties, we
normalize the temperature using the absolute scale, Kelvin.
• As a rule of thumb, the ratio of the environmental temperature, T, to
the melting point, Tm must be less than ~ 0.3 – 0.4, ie.,
T/Tm < 0.3
in order to neglect temperature effects on the mechanical material
properties. Special creep resistant alloys can handle higher T ratios.
Thermal Properties - Glasses
The Glass Transition Temperature, Tg
• This is a property of noncrystalline or amorphous materials.
• If a noncrystalline structure forms, there is a gradual change in
property below the melting point where the material is considered
a supercooled liquid, which is a solid.
• The glass transition temperature applies to glasses, polymers and
more recently, amorphous metals.
General property behaviour of
glass transition temperature, Tg,
for amorphous solids relative to
the melting point, Tm, for
crystalline solids.
When silica crystallizes on cooling there is an abrupt change in
density whereas for the solidification of glass there is not a fixed
temperature.
Glasses
Thermal
Properties
Glass Transition Temperature
The most important noncrystalline materials today are glasses.
A glass is a solid material that has hardened to become rigid without
crystallizing.
The amorphous materials must cool at a fast enough rate to avoid
crystallization. This was very difficult to achieve for the amorphous
metals..
Material Properties for Design of
Structures and Components
Coefficient of Linear Thermal Expansion
• This can also be related to the binding energy between atoms.
• The higher the bonding energy, the lower its thermal
expansion for a given increase in temperature.
• The coefficient of linear thermal expansion, a, is defined
macroscopically as,
dL 1
a
dT L
Where L is the linear dimension of the body.
It is also related to some physical properties such as the specific heat
at constant volume, Cv, and volumetric specific heat at constant
volume, Cv, and the elastic modulus, E,
Where G is the Gruneisen’s
 G  Cv
a
constant, which varies between 0.4
3E
to 4 and is ~1 for metals.
Material Properties for Design of
Structures and Components
Coefficient of Linear Thermal Expansion, a
• Thus a is proportional to 1/E.
• Diamond, which has the highest elastic modulus and one of the
lowest coefficients of thermal expansion, whereas, elastomers,
which have low E, will expand the most.
• From the previous relationships, we see that a is also inversely
proportional to the absolute melting point by,
a
G
100 Tm
For all solids, the thermal strain just before they melt is about the
same.
Examples
Coefficient of Linear Thermal Expansion, a
• In structural materials such as the steels, engineers take into
account a by incorporating expansion joints in structures used for
buildings and bridges.
• In composites such as NiAl/TiB2, the a of the constituent materials
need to be matched very closely to reduce the risk of
delamination at the interfaces due to thermal stresses induced by
thermal cycling.
• In the semiconducting industry, electronic devices are composed of
multiple layers of different materials such as GaAs/AlGaAS,
which can generate defects at the interfaces during thermal
cycling. These defects can destroy or reduce the working lifetime
of the device.
Examples
Opto-electronic materails used for
solar panels, lasers, LEDs, etc.
GaAs will expand or contract more
than Silicon for a unit change in
temperature.
The stress in the epi-layers will be
different during the crystal growth
temperatures (500o – 600 oC) than
when cooled to room temperature.
For device manufacturing, the
residual strain adds another
variable to consider when
manufacturing these material
devices.
An inteface between Silicon and GaAs, showing that the lattice
mismatch that has resulted in the generation of dislocations at
their interface, which penetrate the GaAs and render it useless as
an electronic or photonic material.
Defects in GaAs/InGaAs Laser
Note that the laser looks to have a perfect crystal structure when using
the 200[011] electron diffraction vector but actually has dislocations
when viewed using 220[110].
Dark lines are InGaAs
layers and light lines are
GaAs layers plus
substrate and capping
layers.
Material Properties for Design of
Structures and Components
Thermal Conductivity, k
• In heat conduction, thermal conductivity is the analogous property
to electrical conductivity, s.
• When a temperature gradient, T/x is imposed on a material, the
flux of heat flow per unit area per unit time, Q, is
T
Q  k
x
• where k is in watts per meter per Kelvin.
The analogous equation for the conduction of current density, J, is
V
J s E s
x
Where E is the electric field, V is volts and s.
Material Properties for Design of
Structures and Components
Thermal Conductivity
• Electrons are the principle conductors of heat in materials.
There is a relationship between k and s in metals given as,
k
9 cal ohm
 L  5.5 10
sT
s K 2
• where L is the Lorentz constant.
Thermal conductivity is thus dependent on how fast electrons can
move through the material and how much they scatter off atoms. The
relationship is given by

k

Cvl
Where C is the electron specific heat per unit volume, n is the
electron velocity (2 x 10exp5 m/s), and l is the electron mean free
path, ~10exp-7 m or ~100 nm. What would happen if we decreased
the size of an electronic device to less than l?
Material Properties for Design of
Structures and Components
Thermal Conductivity
• If the mean free path of the electron increases, it’s thermal
conductivity increases.
• Thus superconductors, which have an almost infinitely long mean
free path are very good heat conductors.
• Substituting solid solutes and other atomic-scale defects in metals
will scatter electrons, reduce the mean free path, and thus reduce
thermal conductivity.
Material Properties for Design of
Structures and Components
Thermal Conductivity, k
• In ceramics and polymers, heat conduction is carried by
phonons, not electrons, since electrons are not mobile.
• Phonons are atomic lattice vibrations, which are sensitive to
impurities, lattice defects and surfaces.
• For ceramics and polymers, k is now given by

k  C p vl

Where n is the elastic wave speed (~10exp(3) m/s, which is 2 orders
of magnitude less than the electron velocity) and  is the density and
Cp is the volumetric specific heat.
Material Properties for Design of
Structures and Components
Thermal Conductivity
In microgravity experiments involving plasma crystals conducted in
space, phonons can be studied to help understand physical
phenomena such as heat transfer.
Laser-excited Phonons in a Plasma Crystal
A. Melzer et al. 2001
Phonons
Physical properties
of phonons are being
studied at UVic using
electron holography
(my research).
• The phonons
physical properties of
interest include their
degree of coherence,
spatial coherence
width, and impact
parameter (Coulomb
interaction length).
• What are these?
Ar, 1.2 Pa, 100 W, 8.9 m
Phonons
• The degree of coherence is the fraction of phonons that can interact
with other phonons to create regions of maximum and minimum
intensity regions (hot spots).
• The spatial coherence width is the physical dimension of a phonon,
which is created at the atomic scale and finishes at the millimeter
scale.
• The impact parameter (Coulomb interaction length) is the ability
of phonons to interact with other sources of energy such as another
phonon, an electron or hole, or a crystal defect.
• When an electron flows through a material such as in an electronic
device, it caries with it a phonon.
• When a dislocation moves due to plastic deformation, a phonon
travels down the dislocation’s core.
Phonons And Superconducting Fluxons
Fluxons are small magnets in
superconducting materials
B  B  d S
A
created by two revolving
electrons coupled by phonons.
The phonons act as
the “glue” to bind the
two electrons, known
as a Cooper pair.
The electron creates a
phonon or lattice
distortion in its path.
The phonon is
associated with a net
positive charge.
The other electron is attracted to the net positive charge and
visa versa. The two electrons become bound or coupled.
The two phonons associated with the two electrons require a high degree of
coherence in order to establish a Cooper pair and thus superconductivity.

Superconducting Fluxons
Example of a Superconducting Material, MgB2
The 0 0 0 ½ plane of the hexagonal crystal is occupied by Boron
atoms and has a high density of electrons, which are most likely
supporting the formation of Cooper pair electrons responsible for
the observed superconductivity. This hypothesis needs to be proven,
which may be possible by measuring the degree of coherence of the
phonons existing on this atomic plane.
Canfield and Bud’ko, Scientific America (April 2005) 81.
Material Properties for Design of
Structures and Components
Thermal Conductivity
• If the phonon conductivity is high in ceramics and polymers, then the
thermal conductivity is high.
• Ceramic materials such as diamond, SiC and alumina have this
property (high thermal conductivity) when they are cooled below a
characteristic temperature called the Debye temperature.
• Diamond, SiC and alumina are being used for electronic devices
operating at higher temperatures.
• Amorphous glass has poor thermal conductivity due to its irregular
amorphous structure. It is used as a thermal and electrical insulator.
• Polymers have poor thermal conductivity because the elastic wave
speed n is low and the mean free path of the disordered structure is
small.
Material Properties for Design of
Structures and Components
Thermal Conductivity
• The best thermal insulators are porous materials such as bricks
(firebricks), cork and foams.
• Their conductivity is limited by the gas phase in their cells and by the
heat transfer by radiation through the transparent walls.
• The walls of porous materials that can be coated with a metal such as
Al or Au will reflect the heat wave and thus improve their thermal
insulation.
• NASA uses porous Silica as its heat shield on the Space Shuttle,
which ablates when it melts exposing fresh porous Silica for
insulation.
• We will discuss this more towards the end of the course in the
photonic materials lecture.
The End
(Any questions or comments?)